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4
26
06
Now triangular so determinant is
product of main diagonal entries
=
3
00
1
=−
24.
An example application of determinants is Cramer's rule for solving small systems of
linear equations, described in the next section.
Determinants can also be evaluated through a process known as expansion by
cofactors . First define the minor to be the determinant m ij of the matrix obtained by
deleting row i and column j from matrix A . The cofactor c ij is then given by
1) i + j m ij .
c ij =
(
The determinant of A can now be expressed as
n
n
|
A
| =
a rj c rj =
a ik c ik ,
j
=
1
i
=
1
where r and k correspond to an arbitrary row or column index. For example, given
the same matrix A as before,
|
A
|
can now be evaluated as, say,
4
2)
+
6
=−
4
26
250
50
1
20
25
|
A
| =
=
(
80
16
+
72
=−
24.
4
2
4
21
21
4
3.1.4 Solving Small Systems of Linear Equation Using
Cramer's Rule
Consider a system of two linear equations in the two unknowns x and y :
ax
+
by
=
e , and
cx
+
dy
=
f .
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